Abstract—To understand the dynamics and health of marineecosystems such as lagoons and coral reefs as well as tounderstand the impact of human activities on these systems, itis imperative to predict the residence times of water masses andconnectivity between ocean domains. In the present work, weconsider the pristine lagoons and coral reefs of the Red Sea asan example of such sensitive ecosystems, with a large numberof marine species, many of which are unique to the region. Tostudy the residence times and connectivity patterns, we makeuse of recent advances in dynamic three-dimensional Lagrangiananalyses using partial differential equations. Specifically, weextend and apply our novel efficient flow map compositionscheme to predict the time needed for any particular waterparcel to leave the domain of interest (i.e. a lagoon) as well asthe time for any particular water parcel to enter that domain.These spatiotemporal residence time fields along with four-dimensional Lagrangian metrics such as finite time Lyapunovexponent (FTLE) fields provide a quantitative description of theLagrangian pathways and connectivity patterns of lagoons in theRed Sea.
Index Terms—Red Sea, Lagoons, Ocean modeling, Lagrangianfield analysis, Residence time, Flow map composition, FTLE,Lagrangian coherent structures
I. INTRODUCTION
Predicting the residence times and biophysical connectivity
of ocean regions is extremely important to characterize the
behaviors, dynamics, and health of marine ecosystems as well
as to predict the effects of human activities in localized areas
on the ecosystems connected to these areas. This is especially
critical for the health and resiliency of marine lagoons and
coral reefs. Considering the case of the Red Sea, its lagoons
and coral reefs constitute an amazing undersea world home
to 300 hard coral species and about 1,200 fish, of which 10
percent are local to the region. Its large number of lagoons
along the coast (75 on the coast of Saudi Arabia) have large
residence times that help coral growth due to the absence of
erosion. However, the restricted exchange of water between
these lagoons and the Red Sea is also responsible for pollution
in the lagoons [19]. Characterizing the different connectivities
in the region is thus very important. For example, [18]
have showed that connectivity patterns can explain the gene
diversity of the coral reefs found in the region.
To understand the connectivity patterns and study the ex-
change of water masses between various lagoons in the Red
Sea, we resort to using Lagrangian field analyses. In the
broadest sense, such analyses refer to studies performed using
the Lagrangian viewpoint of fluid mechanics [14]. These field
analyses provide a quantitative understanding of the transport
characteristics of passive materials that flow with the fluid.
They compute the attracting basins and repulsive surfaces, and
accurately predict the flow patterns of such passive material
through determining coherent and incoherent sets [4, 8, 13].
Extensive work has been done regarding Lagrangian transport
in geophysical systems, and we refer the readers to [5, 6] for
comprehensive reviews.
To address the challenges in lagoons and coral reefs,
we utilize our recent advances in efficient four-dimensional
(4D: time and space) Lagrangian field theory and methods
[2, 3, 11, 12] for characterizing in a principled fashion
the residence times and connectivity fields, showcasing the
results for the Red Sea. Specifically, we study the connectivity
patterns between the Eastern and Western coasts of the Red
Sea Basin and the isolation of the southern part of the sea. By
looking at how the structures of the flow evolve in presence
of seasonal streams, we better understand the effects of these
streams on connectivity patterns. Our approach is rooted in the
fundamental Eulerian partial differential equations (PDEs) for
the Lagrangian flowmap. With our novel numerical method
of composition, we can solve these PDEs accurately and
efficiently without compounding numerical errors [12]. As a
result, instead of classic trajectory-based analyses, we provide
accurate 4D field characterization of the Red Sea coherent wa-
ter masses, residence time, and connectivity dynamic features.
The long-term goal is to provide sustained 4D Lagrangian pre-
dictions, analyses, and characterizations of multiscale ocean
transports, coherent structures, material sets, residence times,
connectivity, and stirring and mixing processes in the Red Sea
Doshi, M.M., C.S. Kulkarni, W.H. Ali, A. Gupta, P.F.J. Lermusiaux, P. Zhan, I. Hoteit, and O.M. Knio, 2019. Flowmaps and Coherent Sets for Characterizing Residence Times and Connectivity in Lagoons and Coral Reefs: The Case of the Red Sea. In: OCEANS '19 MTS/IEEE Seattle, 27-31 October 2019, doi:10.23919/OCEANS40490.2019.8962643
To represent the unsteady and multiscale nature of the
ocean fields in the Red Sea region, we utilize the MIT
general circulation model (MITgcm) [16] in 3D, including
8 major tidal components. The model is forced by hourly
atmospheric fluxes, and a fine scale bathymetry field generated
by assimilating several in situ observations is employed. The
dynamical characteristics of the regional lagoon basins are
mainly forced by tides at local scales, whereas the long
term circulation inside the lagoons is driven by the general
circulation outside the lagoon. Another motivation of the
present work is to investigate these regional dynamics in the
Lagrangian field sense and, in the future, collaborate with
observational scientists to help design adaptive monitoring
campaigns and plan principled optimal sampling strategies for
characterizing the 4D residence times and connectivity fields
in the lagoons and coral reefs.
The long-term objectives of the present collaborative Red
Sea research are to: (i) Utilize our new Lagrangian field
transport theory and methods to forecast, characterize and
quantify ocean processes involved in the four-dimensional
transports and transformation of water masses, and residence
times in the Red Sea; (ii) Apply and expand our multi-
way nesting, and uncertainty predictions, for real-time fore-
casting and process studies in the region; (iii) Help design field
experiments and predict sampling strategies that maximize
information on residence times, 4D pathways and dynamics
in the region.
The manuscript is organized as follows. Section II
overviews the state-of-the-art in Lagrangian analyses and our
novel field PDE-based approach, with a specific focus on
marine environments. Section III discusses the ocean modeling
methodology and the regional oceanography of the Red Sea.
We then showcase in Section IV selected results regarding the
Lagrangian field dynamics of the Red Sea and its residence
times, with an emphasis on specific lagoons. Finally, conclu-
sions are provided in Section V.
II. LAGRANGIAN METHODS
A. Lagrangian Fields and Computation
We now briefly review the fundamentals of the Lagrangian
viewpoint of material transport field analysis, and the associ-
ated recent advances. Typically, we denote any quantity that
is being passively advected by the background fluid flow as
a (passive) tracer, denoted henceforth by α(x, t), where x is
the (vectorial) position in the domain of interest Ω and t is
the time, with t ∈ [0, T ]. Examples of typical passive tracers
include temperature, salinity, inertia-free particulate matters
etc. We assume that the tracer quantity α(x0, t0) that was
at location x0 at time t0 is passively transported with the
underlying fluid parcel that was at location x0 at time t0, and
ends up at location x at time t. Thus, we have that:
α(x, t) = α(x0, t0) = α0(x0) (1)
However, we know that the motion of the fluid parcel is
governed by eq. 2.
x(t) = v(x(t), t) given x(t0) = x0 (2)
where v(x, t) is the dynamic velocity field in the domain Ω.
For the dynamical system given by eq. 2, the forward flow
map between times t0 and t1(≥ t0) is defined as:
φtt0(x0) = x where x(t) = v(x(t), t) with x(t0) = x0 (3)
That is, the forward flow map is simply the position of the fluid
parcel at some later time (t) mapped onto its initial position
(at time t0). The inverse of the forward flow map, called the
backward flow map is analogously given by eq. 4, where now
the transport ODE 2 is solved in backward time with a specific
terminal condition:
φtt0(x) = x0 where x(t) = v(x(t), t) with x(t) = x (4)
Substituting eq. 4 into eq. 1, we obtain eq. 5 that concisely
states
α(x, t) = α0(φtt0(x)) . (5)
Eq. 5 suggests that computing a passive tracer transport ul-
timately amounts to accurately computing the flow maps of
the underlying dynamical system and composing the said flow
maps with the tracer initial condition.
The forward and backward flow map fields also provide
a wealth of additional information. The singular values of
the Jacobians of these maps, when scaled logarithmically are
referred to as the ’finite time Lyapunov exponents’ (FTLEs)
[7, 17]. These forward and backward FTLE fields are com-
monly used to identify Lagrangian coherent structures (LCSs).
The ridges of the forward FTLEs approximate the repelling
manifolds: they tend to ’repel’ water parcels. Two parcels that
are close to each other at initial time but on different sides of
the forward FTLE ridge will tend to advect farther away from
each other than other parcels. The forward FTLEs thus act as
material barriers to connectivity, and the forward FTLE ridges
can be thought of as a skeleton to the connectivity pattern. On
the other hand, the ridges of the backward FTLEs approximate
the attracting manifolds: they tend to ’attract’ water parcels.
They thus increase the chances of connectivity among different
water regions, ultimately by sub-mesoscale or turbulent mixing
along the ridges in the backward FTLEs. Several other theories
and metrics rooted in the flow map are used to determine
attracting - repelling manifolds, coherent - incoherent material
sets and other quantities of interest in fluid flows [4, 6, 8].
The typical trajectory-based approach to compute the flow
maps is to appropriately solve eq. 2 in forward or backward
time using certain time marching schemes for all possible
initial conditions. However, the same can also be achieved
by solving a single PDE whose characteristics are described
by the said ODE. Specifically, one can obtain the backward
flow map φ0t by solving the PDE eq. 6 forward in time from
time 0 to t, with the initial condition α0(x) = x:
∂α
∂t+ v · ∇α = 0; α0(x) = x then α(x, t) = φ0
t (x) . (6)
Similarly, the forward flow map φt0 is obtained by solving eq. 7
backward in time, with the terminal condition αt(x) = x:
∂α
∂t+ v · ∇α = 0; αt(x) = x then α(x, 0) = φt
0(x) . (7)
Once the flow map is computed, its associated quantities
can be appropriately computed. Further, the flow map can
be composed with the tracer initial condition to obtain the
advected tracer field.
Finally, instead of computing the flow maps over the entire
considered interval, one can also compute flow maps over
smaller intervals and then compose them appropriately to
obtain the flow maps over the larger time interval. Specifically:
φtnt0 = φtn
tn−1◦ φtn−1
tn−2◦ . . . φ2
1 ◦ φ10 (8)
φt0tn = φ0
1 ◦ φ12 ◦ . . . φtn−2
tn−1◦ φtn−1
tn (9)
We refer to this method are the ‘method of flow map compo-
sition’. Composing such independent flow maps over smaller
intervals presents the opportunity to parallelize the computa-
tion in the temporal direction, yielding a significant speedup.
The individual flow maps are computed over a short interval
and hence introduce minimal numerical errors. Further, the
individual flow map computations are independent and hence
the numerical errors are not compounded, which results in a
much lower total error. Further details can be found in [12].
B. Residence Times Fields
Presently, we are primarily interested in efficiently com-
puting the residence time fields in the domain of interest.
These fields represent the time required for a water parcel that
started at the considered position to leave the chosen domain
of interest. An efficient approach to compute these fields is
now developed.
Let ΩD be the domain of interest in which we wish to
compute the residence times. We initialize a hypothetical tracer
field given by:
α(x) =
{1 ∀x ∈ ΩD
0 otherwise. (10)
As stated earlier (eq. 5), the tracer concentration at a location
x at some time t, i.e. α(x, t) is computed using the initial
tracer concentration (α0(x)) as:
α(x, ti) = α(φ0ti(x)
). (11)
This equation can simply be inverted to obtain eq. 12.
α0(x) = α(φti0 (x), ti
). (12)
We can now use this initial concentration field to determine
which water parcels (with their positions indexed at the initial
time t = 0) will be inside or outside the domain of interest at
t = ti, specifically:
1) α0(x) = 0 and x ∈ ΩD (inside-outside): Water parcel
at x starts in the domain of interests and is outside the
domain at time ti.
2) α0(x) = 1 and x ∈ ΩD (inside-inside): Water parcel at xstarts in the domain of interests and stays in the domain
at time ti.3) α0(x) = 0 and x �∈ ΩD (outside-outside): Water parcel
at x starts from outside the domain and is outside the
domain at time ti.4) α0(x) = 1 and x �∈ ΩD (outside-inside): Water parcel at
x starts from outside the domain and is inside the domain
at time ti.
By computing these values at different times ti we can
construct the residence time fields based on when a water
parcel exits the domain. Similar fields can be constructed to
represent the intruding time fields as well as the other two
cases described above.
III. OCEAN MODELING AND REGIONAL DYNAMICS
A high-resolution MIT general circulation model (MITgcm)
[16] was configured at KAUST for the coastal lagoon area
offshore of Al Wajh region in the central Red Sea, extending
from 36.35◦E to 37.25◦E and from 25.2◦N to 26.4◦N (Figure
1). The coastal model has a horizontal resolution of about 75
m and 50 vertical z-levels, the thickness of which gradually
increases from 0.5 m at the surface to 180 m near the bottom.
The model is nested within a regional 1-km model configured
for the Red Sea with temperature, salinity, and horizontal
velocity fields prescribed at the southern and western bound-
aries on a daily basis. Barotropic tidal currents are added
through the amplitudes and phases extracted from the inverse
barotropic tidal model TPXO 7.2 Indian Ocean (Red Sea)
[REF], including eight major tidal components of semi-diurnal
and diurnal frequencies (M2, S2, N2, K2, K1, O1, P1 and Q1).
The model is forced by hourly surface wind, air temperature,
specific humidity, precipitation, and downward shortwave and
longwave radiation of a 3-km Weather Research Forecast
(WRF) product [20] downscaled from ERAInterim products
of the European Centre for Medium-Range Weather Forecasts
(ECMWF) [1]. Various sources of data have been collected and
merged to generate the fine-scale model bathymetry, including
several in-situ cruise measurements that were conducted in
the lagoon, the General Bathymetric Chart of the Oceans
(GEBCO) [9] and satellite images from Google Earth.
At local scales in the Al Wajh region, tides constitute the
major forces that determine the dynamical characteristics of
lagoon-like basins; however, the long-term circulation inside
the lagoons is primarily driven by larger-scale meteorological
forces and the general circulation outside the lagoon. North-
westerly winds are dominant over the sea all over the year [15],
while the wind regime from the land is more variable with
smaller valleys cutting across the mountain ridges and lead
to strong easterly jets [10]. The regional oceanic circulation
commonly consists of a northward boundary current along
the Saudi coast [21, 22], and frequent eddies that are more
energetic during winter [23, 24, 25, 26]. Such eddy and
boundary current events may affect the regional circulation
outside the lagoon and potentially influence the flow inside.
Fig. 1: Modelling Domains
IV. LAGRANGIAN DYNAMICS AND RESIDENCE TIMES
RESULTS
We now apply the above methodology to compute flow
maps, FTLEs, and residence times in the Al Wajh region
and overall Red Sea. We then showcase how the FTLE fields
qualitatively depicts the skeleton of connectivity in these
regions.
Figures 2 show the forward x and y flow maps at 0.25 m
and Figure 3 the forward x, y and z flow maps 2.75 m, all
plotted at the initial positions. That is, these maps collectively
plot the final position (through x, y, and z coordinates, with
z being positive downward) of a water parcel that started at a
particular location at the start time. From the intensity in the
gradients of the flow map, it can be seen that the amount of
mixing in the Lagoon is much less than outside the Lagoon.
Figure 4 shows the vertical sections of the 3D x, y and zflow maps, with the Red Sea located to the left of the shown
sections. The tall ridges in the repelling FTLE field signify
very little mixing with the Red Sea. The z flow map shows
upwelling at the eastern ridge of the Lagoon and downwelling
on the western coast of the Lagoon. Hence, the regions of
larger vertical transport within the Lagoon are located near its
edges. Nutrient inputs can thus be expected to occur there as
well.
The forward FTLE fields computed using the flow maps
from Figure 2 are shown in Figure 5. We illustrate the forward
FTLE for two different depths: 0.25 m (i.e. close to the surface)
and 2.75 m from Nov. 3, 2017 to Nov. 10, 2017. As mentioned
before, forward FTLEs tend to repel water parcels and thus act
as barriers to connectivity. In Figures 5(a) and (b), we see a
prominent FTLE ridge on the north side of the domain at both
the considered depths. Further, we also highlight the existence
of the ridges connecting the different small islands at the
southwestern boundary of the lagoon. As the forward FTLEs
form barriers to connectivity, these ridges together imply that
during the 7-days considered, the water in the lagoon does not
mix very well with the waters offshore at the southwestern
boundary. This can be further confirmed by the FTLE fields
over the entire red sea in Figure 6 (from Jan. 1, 2006 to Jan.
13, 2006), where we see prominent FTLE ridges between the
Red Sea and the northeastern coastal regions. Moreover, there
seems to be a significant amount of mixing of parts of the
lagoon waters and the Red Sea on the northwest and southeast
boundaries of the Al Wajh lagoon.
While the FTLE fields can indicate which waters of the
lagoon are most likely (or not likely) to mix with the offshore
Red Sea waters, they do not tell us directly where the mixing
occurs spatially (i.e. whether the sea water enters the lagoon
or whether the lagoon water enters the offshore sea). The
aforementioned residence time fields can be utilized to perform
this analysis.
To this end, Figure 7 shows residence time fields for water
parcels inside the lagoon. Specifically, what is show is the
amount of time it takes for a water parcel that started at the
specific position to leave the lagoon. We observe that waters
near the southeastern boundary exit the domain while those on
the northwestern boundary remain inside the lagoon. We also
see that the parcels that leave the domain are not uniformly
spread out in the lagoon but form a distinct structure near
the south eastern boundary. This egress happens mainly at
the surface as the residence time fields at 2.75 m depth show
significantly less water exiting the domain, see Figure 7(b).
This is also verified by the stronger FTLE ridges at 2.75 m,
see Figure 5(b), at the border of the lagoons implying lower
horizontal mixing at depth.
Using our PDE-based Lagrangian methodology, we can also
compute when waters from outside the Lagoon in the offshore
Red Sea have entered the Lagoon. We refer to these fields
as the “Entrance time” fields, as shown in Figure 8. We see
that the outside waters enter the northwestern boundary of
the Lagoon mainly at the surface. This is the portion of the
offshore sea which we had predicted would mix with the
waters in the Lagoon. We also note that the seawater parcels
that enter the Lagoon form interesting filament structures, as
seen in Figure 8.
Finally, we observe structures on the southwestern boundary
of the Lagoon that enter the domain of interest. Since we
predict that these waters in the Red Sea will not significantly
mix with the Lagoon waters, we predict that most of these
waters enter and then leave the lagoon domain during the
period of interest, without significant mixing.
V. CONCLUSIONS
In this work, we studied the residence times and con-
nectivity patterns of water masses in the Red Sea and Al
Wajh Lagoon region. Such capabilities are of great use in
characterizing the behavior, dynamics and health of the marine
ecosystems native to the region.
To compute the residence times and connectivity patterns,
we resort to recent advances in efficient four-dimensional
(a) x flow map (b) y flow map
Fig. 2: Forward flow maps (m) over the Al Wajh Lagoon modeling domain from 3 Nov, 2017 to 10 Nov, 2017 at 0.25 m.
(a) x flow map (b) y flow map (c) z flow map
Fig. 3: Forward flow maps (m) over the Al Wajh Lagoon modeling domain from 3 Nov, 2017 to 10 Nov, 2017 at 2.75 m (zbeing positive downward).
(3D+time) Lagrangian analyses using partial differential equa-
tions. Specifically, we show how the method of composition
can be efficiently used to compute the residence time fields,
i.e. spatial fields that denote the amount of time a water parcel
spends before exiting the domain of interest. With the same
method, we also compute and describe the entrance time fields,
(a) x flow map (b) y flow map (c) z flow map
Fig. 4: Vertical section of forward flow maps over the Al Wajh Lagoon modeling domain from 3 Nov, 2017 to 10 Nov, 2017
at 25.61◦N.
(a) forward FTLE at 0.25 m (b) forward FTLE at 2.75 m
Fig. 5: Forward FTLEs over the Al Wajh Lagoon modeling domain, from Nov. 3, 2017 to Nov. 10, 2017 at 0.25 m and 2.75 m,
respectively.
Fig. 6: Forward FTLEs over the entire Red Sea from Jan. 1,
2006 to Jan. 13, 2006.
i.e. spatial fields that denote the amount of time before water
parcels offshore enter the domain of interest. We confirm that
the ridges of the forward FTLE fields (approximating the
repelling coherent structures) do indeed correspond to barriers
to material flow, and we observe minimal water flux across
these FTLE ridges in the simulations. That is, the residence
time for most of the waters inside the Al Wajh Lagoon region
is large, and a comparatively little amount of water mass leaves
the Lagoon region. This indicates that even though the Lagoon
waters are physically connected to the larger Red Sea offshore,
they are only weakly connected in terms of material flow.
This is especially important as the Lagoon ecosystem remains
relatively protected from any major disturbances or events in
the Red Sea. We believe that such in-depth scientific analyses
of the biogeochemical ecosystems and their connectivities
would go a long way in making informed policy decisions
regarding their conservation.
ACKNOWLEDGMENTS
The study was supported by King Abdullah University of
Science and Technology (KAUST) under the “Virtual Red
(a) Residence Time at 0.25 m (b) Residence Time at 2.75 m
Fig. 7: Residence time fields in the Al Wajh region, from Nov. 3, 2017 to Nov. 10, 2017 at 0.25 m and 2.75 m, respectively.
(a) Entrance Time at 0.25 m (b) Entrance Time at 2.75 m
Fig. 8: Entrance time fields in the Al Wajh region, from Nov. 3, 2017 to Nov. 10, 2017 at 0.25 m and 2.75 m, respectively.
Sea Initiative” award REP/1/32680101 and made use of the
resources of the Supercomputing Laboratory and computer
clusters at KAUST. The MSEAS group is also grateful
to the Office of Naval Research (ONR) for support un-
der grants N00014-14-1-0725 (Bays-DA), N00014-18-1-2781
(DRI-CALYPSO), and N00014-19-1-2693 (IN-BDA), and to
the National Science Foundation for support under grant EAR-
1520825 (NSF-ALPHA), all to the Massachusetts Institute of
Technology.
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